Degenerate (r−2)-dimensional subvarieties through the generic hyperplane section of a reduced irreducible complex curve in Pr

We prove the following result: the generic degenerate (r−2)-dimensional subvariety through the generic hyperplane section of a complex reduced irreducible curve inP ^r is smooth at each point of the section     

Fast computation of a rational point of a variety over a finite field

We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time-space complexity is roughly quadratic in the logarithm of the cardinality of the field and a geometric invariant of the input system. This invariant, called the degree, is bounded by the Bézout number of the system. Our algorithm works for fields of any characteristic, but requires the cardinality of the field to be greater than a quantity which is roughly the fourth power of the degree of the input variety.

     

The index of an algebraic variety

Let K be the field of fractions of a Henselian discrete valuation ring  ${{\mathcal {O}}_{K}}$ . Let X _K /K be a smooth proper geometrically connected scheme admitting a regular model $X/{{\mathcal {O}}_{K}}$ . We show that the index δ(X _K /K) of X _K /K can be explicitly computed using data pertaining only to the special fiber X _k /k of the model X. We give two proofs of this theorem, using two moving lemmas. One moving lemma pertains to horizontal 1-cycles on a regular projective scheme X over the spectrum of a semi-local Dedekind domain, and the second moving lemma can be applied to 0-cycles on an $\operatorname {FA} $ -scheme X which need not be regular. The study of the local algebra needed to prove these moving lemmas led us to introduce an invariant γ(A) of a singular local ring $(A, {\mathfrak {m}})$ : the greatest common divisor of all the Hilbert-Samuel multiplicities e(Q,A), over all ${\mathfrak {m}}$ -primary ideals Q in ${\mathfrak {m}}$ . We relate this invariant γ(A) to the index of the exceptional divisor in a resolution of the singularity of $\operatorname {Spec}A$ , and we give a new way of computing the index of a smooth subvariety X/K of ${\mathbb{P}}^{n}_{K}$ over any field K, using the invariant γ of the local ring at the vertex of a cone over X.     

The general hyperplane section of a curve

In this paper, we discuss some necessary and sufficient conditions for a curve to be arithmetically Cohen-Macaulay, in terms of its general hyperplane section. We obtain a characterization of the degree matrices that can occur for points in the plane that are the general plane section of a non-arithmetically Cohen-Macaulay curve of . We prove that almost all the degree matrices with positive subdiagonal that occur for the general plane section of a non-arithmetically Cohen-Macaulay curve of , arise also as degree matrices of some smooth, integral, non-arithmetically Cohen-Macaulay curve, and we characterize the exceptions. We give a necessary condition on the graded Betti numbers of the general plane section of an arithmetically Buchsbaum (non-arithmetically Cohen-Macaulay) curve in . For curves in , we show that any set of Betti numbers that satisfies that condition can be realized as the Betti numbers of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay curve. We also show that the matrices that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, integral, (smooth) non-arithmetically Cohen-Macaulay space curve are exactly those that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay space curve and have positive subdiagonal. We also prove some bounds on the dimension of the deficiency module of an arithmetically Buchsbaum space curve in terms of the degree matrix of the general plane section of the curve, and we prove that they are sharp.

     

On a hyperplane section theorem of Gurjar

A hyperplane section theorem by R. V. Gurjar [3] is given a short proof. Power series in any number (at least three) of variables satsifying the condition of the theorem are explicitly constructed. In the course of the proof, the restrictive-looking condition of the theorem is given an easy sufficient condition from the view point of the weighted projective spaces. Received April 12, 2000 / Accepted August 21, 2000 / Published online October 30, 2000     

Generic section of a hyperplane arrangement and twisted Hurewicz maps

We consider a twisted version of the Hurewicz map on the complement of a hyperplane arrangement. The purpose of this paper is to prove surjectivity of the twisted Hurewicz map under some genericity conditions. As a corollary, we also prove that a generic section of the complement of a hyperplane arrangement has nontrivial homotopy groups.     

Conditions on decomposable symmetric tensors as an algebraic variety

Let V be a finite dimensional vector space of dimension at least 2 over an infinite field F. We show that the set of all decomposable elements in the rth symmetric product space over i:V(r≥ 2) is an algebraic set if F is algebraically closed and only if every polynomial of degree at most r splits completcly over F.     

Conditions on decomposable symmetric tensors as an algebraic variety

Let V be a finite dimensional vector space of dimension at least 2 over an infinite field F. We show that the set of all decomposable elements in the rth symmetric product space over i:V(r≥ 2) is an algebraic set if F is algebraically closed and only if every polynomial of degree at most r splits completcly over F.     

On the genus of a hyperplane section of a geometrically ruled surface

Summary In this paper we estimate the minimal genus of hyperplane sections of a geometrically ruled surface.